The Math Forum

Ask Dr. Math - Questions and Answers from our Archives
Associated Topics || Dr. Math Home || Search Dr. Math

Limits of Sequences

Date: 02/25/2001 at 02:11:06
From: N. D. Kapoor
Subject: A question about limits

Dear Dr. Math,

We know that the limit (n -> infinity) [(1 + 1/n)^n] = e. Is the limit 
(n -> infinity) [(1 + 1/sqrt(n))^(1.5n)] also e? Also, what is the 
limit (n -> infinity) [(1 + a/n)^n], where a is not equal to 0?

Please help.
N. D. Kapoor

Date: 02/26/2001 at 22:27:50
From: Doctor Fenton
Subject: Re: A question about limits

Dear Mr. Kapoor,

Thanks for writing to Dr. Math. For questions like these, I like to 
use the fact that for a > 0 and b real,

     a^b = e^[b*ln(a)]

where ln is the natural logarithm. Then:

     (1 + 1/sqrt(n))^(1.5n) = e^[1.5*n*ln(1 + 1/sqrt(n))]

The exponent can be written as the indeterminate form:

     ln( 1 + 1/sqrt(n))

By L'Hospital's Rule, the limit of this expression as n->oo is the 
same as the limit as n->oo of:

           1            1
      ----------- * ---------
      1 + sqrt(n)   2*sqrt(n)

and the quotient is of order n as n->oo, so the expression diverges to 
-oo, which means the original expression -> 0, not e.

For the second expression, you can repeat this computation, or write:

     (1+a/n)^n = (1 + a/n)^[(n/a)*a]

               = [(1 + a/n)^(n/a)]*a

               = {[ 1 + 1/(n/a)]^(n/a)}^a 

and the term in braces {} approaches e by your earlier observation.

If you have further questions, please write us again.

- Doctor Fenton, The Math Forum   
Associated Topics:
High School Calculus
High School Sequences, Series

Search the Dr. Math Library:

Find items containing (put spaces between keywords):
Click only once for faster results:

[ Choose "whole words" when searching for a word like age.]

all keywords, in any order at least one, that exact phrase
parts of words whole words

Submit your own question to Dr. Math

[Privacy Policy] [Terms of Use]

Math Forum Home || Math Library || Quick Reference || Math Forum Search

Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.